Homology:

Intuition of Homology: is some dimensional object, and gets its boundary. suggests a boundary’s boundary is . is the dimensional things that no more has a boundary. is the dimensional things that is a boundary (of some dimensional thing). Now from this, it is still hard to visualize what is. This suggests that homology does not have a general intuition.

Instead, let’s look at the simplest example in simplicial homology: a labeled by its three vertices, . We have , , thus . On the other hand, if we remove the face leaving only the skeleton, , thus . We see that is like “the boundary of something”, and $latex \text{Im }\partial_n$ is like “is this ‘something’ filled or not”? if it is filled, then we have no hole. if it is not filled, then we have a hole. This hole is captured by . However, this intuition is only true for the part of the homology (one corresponds to one hole) and has not accounted for (e.g.) being a homology group.

In fact, finite part shows the degree of non-triviality a hole is. If it is just an ordinary hole, we can wrap it around as many times as we want so we get a . However, if a hole is nontrivial meaning some wrapping actually deforms to identity (no wrapping at all), then we get some finite part like . An example would be .

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